Eyeglass lens designing method and eyeglass lens

ABSTRACT

It is made possible to easily obtain a spectacle lens with higher performance in a spectacle lens designing method in which an eyeball motion (Listing&#39;s Law) is taken into consideration. A spectacle lens designing method in which an eye motion (Listing&#39;s Law) is taken into consideration, and which uses, as an evaluation function regarding visual acuity constituting a merit function which is used in optimization calculation, a visual acuity evaluation function (log MAR) derived in an ordinary manner from a visual acuity measured value V which is actually measured. Note that the visual acuity evaluation (log MAR) is represented by the following equation (1), letting a curvature of field be an ordinary aberration of a spectacle lens, and a residual astigmatism be an astigmatism extendedly defined from the spectacle lens designing in which the Listing&#39;s Law is taken into consideration.  
     log  MAR =log 10  (1/ V (curvature of field, residual astigmatism))

TECHNICAL FIELD

[0001] The present invention relates to a spectacle lens designingmethod and a spectacle lens designed by the same.

BACKGROUND ART

[0002] The Listing's Law in an eyeball motion means that, when aneyeball looks far forward (first eye position), a rotation axis of theeyeball motion exists in a surface including the center of rotation ofthe eyeball and being perpendicular to this eye position (Listing'ssurface). In this case, when the eyeball rotates from the first eyeposition along spectacle principal meridians (representing two verticaland horizontal lines orthogonal to each other on a Gaussian curvedsurface and representing the same below) according to the Listing's Lawat the time one wears astigmatic spectacles, the spectacle principalmeridians and axes of a coordinate system rotating according to theListing's Law become parallel to each other and an angle between thembecomes 0.

[0003] However, when the eyeball motion changes in a direction differentfrom the spectacle meridians, the angle made by the spectacle meridiansand the coordinate axes rotating according to the Listing's Law do notbecome 0 to cause an angle deviation.

[0004] By taking this angle deviation of the coordinate system intoconsideration, an accurate astigmatism and curvature of field (alsocalled a power error) can be calculated.

[0005] A spectacle lens designing method in which this eyeball motion(Listing's Law) is taken into consideration is disclosed in JapanesePatent Laid-open No. Sho 57-10112 (hereinafter, referred to as Prior art1)(refer to FIG. 5 in Prior art 1).

[0006] Meanwhile, optimization of evaluation functions for several kindsof aberrations, a lens shape, and so on by optimization calculation inan aberration correction process in designing a lens is known as isdisclosed, for example, in Japanese Patent Publication No. Hei 2-38930.

[0007] To explain the outline of this optimization calculation, takingdesigning of a single vision aspherical lens for example, though it is aknown technique in spectacle lens designing, data on a lens material andprescription specifications are given as basic design specifications,items such as a center thickness are further included as additionalspecifications in a case of a positive lens, and a combination ofrefractive surface shapes of a front surface and a rear surface whichsatisfies them and has as small an optical aberration as possible isobtained by calculation. The refractive surface is expressed as asurface which is mathematized by a function and the function consists ofa plurality of parameters defining a spectacle lens. The parametersinclude a refractive index of the material, a lens diameter, radii ofcurvature of the front surface and the rear surface, the centerthickness, an aspherical conic coefficient, a high degree asphericalcoefficient, and so on. They are classified into fixed factors andvariable factors according to the object of the lens designing, and thevariable factors are dealt as variable parameters.

[0008] Then, using a ray tracing method and a wave front tracing method,a plurality of evaluation points whose distances from an optical axis onthe refractive surface are different are set on the lens surface, theoptical aberration on each of the evaluation points is expressed as anevaluation function (merit function), and calculation to obtain theminimum evaluation function is done using an optimization calculationmethod such as a damped least square method. At this time, simulationsare repeated while operating the variable parameters of the refractivesurface, and when a target value is obtained, the final shape of therefractive surface is determined.

[0009] As the parameters constituting the evaluation function (meritfunction) in the optimization calculation, an astigmatism and acurvature of field are generally known, and in a case, for example, whenthe front surface and the rear surface are both spherically designed ina designing method in a prior art, assuming that the aberrationsshowing, in a unit of diopter, two focal positions Ft, Fs obtained bythe ray tracing method based on a focus D obtained by a paraxial raytracing are t (tangential surface) and s (sagittal surface) as shown inFIG. 11, a lens in which the astigmatism=(t−s) is minimum is called aTscherning Form and a lens in which the curvature of field=(t+s)/2 isminimum is called a Percival Form. In Japanese Patent Publication No.Sho 42-9416, an evaluation function in which t and s are complicatedlycombined and which is expressed as a horizontal aberration is disclosed.

[0010] A distortion aberration is known to be also an importantevaluation function in the aforesaid design optimization calculation,and designing in which it is taken into consideration is proposed, forexample, in Japanese Patent Laid-open No. Sho 55-59425 (hereinafter,referred to as Prior art 2) and APPLIED OPTICS, Vol. 21, No.162982-2991: written by Milton Katz (hereinafter, referred to as Priorart 3).

[0011] As one of free curved surfaces among lens refractive surfaceshapes, there is an atoric surface besides a spherical surface and anastigmatic surface, and it is known as a surface expressing a line otherthan the spectacle principal meridians on the astigmatic surface. Theuse of a spline function as an equation used to express the atoricsurface is disclosed in Japanese Patent Laid-open No. Sho 62-30216(Prior art 4) and an equation using orthogonal functions of xy isdisclosed in International Publication No. WO 93/07525 (hereinafter,referred to as Prior art 5) is disclosed.

[0012] In recent years, however, it has been found out that visualacuity is closely related to processing in the brain and it has beenknown that the visual acuity is mainly constituted by an image on aretina and processing of the image in the retina and the brain.

[0013] Meanwhile, in the designing of spectacle lenses in the prior art,such an idea has been dominant that performance of a spectacle lens isimproved as optical performance of the lens becomes higher.

[0014] For example, in the optimization calculation method describedabove, the evaluation function in the prior art is based on anevaluation only by optical calculation, such as evaluation of the sizeof an image and t (tangential surface) and s (sagittal surface) of theaberration and so on which are calculated at a far point sphere (FPS) inFIG. 11 by the ray tracing method, and furthermore, an image plane or aretina surface are also dealt as a film surface of a camera withoutconsidering a physiological function of an eye such as the eyeballmotion.

[0015] Furthermore, since the distortion aberration is dealt as anoptical amount of a camera as described above also in theabove-mentioned Prior art 3, the evaluation function used in it isdifferent from an evaluation function based on a visual anglemagnification M which is used in spectacles (for example, KOHGAKU(OPTICS), Vol. 19, No. 10 “Futatabi Kakubairitsu nitsuite (On AngleMagnification Again)” Kazuo Miyake), and furthermore, an astigmatic lensand the designing in which the eyeball motion is taken intoconsideration are not disclosed either. Furthermore, the above-mentionedPrior art 2 does not disclose any concrete technical content thereof andits actual state is not clear.

[0016] Meanwhile, in lens designing, the use of the spline function forthe atoric surface having a higher degree of freedom of expression,which is disclosed in the above-mentioned Prior art 4, enables theexpression of free surface shapes, but it has a disadvantage that itbasically lacks precision in surface expression. Moreover, in theabove-mentioned Prior art 5, the properties of the eyeball motion usingthe Listing's Law are not utilized to result in an insufficient opticalsurface.

[0017] Prior art 1 discloses a designing method in which the eyeballmotion is taken into consideration using the Listing's Law. However,here, the explanation of the above-described technical idea is focusedon, and in the concrete lens designing, performance evaluation is madebased only on an astigmatism derived from optical calculation, and anevaluation function in the optimization calculation is insufficient.

[0018] Moreover, no concrete disclosure on the expression of a lenssurface is given.

[0019] Furthermore, designing in this Prior art 1 is essentially thesame as the one in the prior art based on the idea that performance of aspectacle lens is improved as optical performance becomes higher and itgives no consideration to the correlation with visual acuity.

[0020] Thus, it is clear that performance evaluation of a spectacle lensbased only on indexes such as an optical amount on the retina and theaberrations is inaccurate as a simulation on a living human body sinceno consideration is given to the viewpoints of the processing in theretina and the brain and of the eyeball motion as described above.

[0021] An object of the present invention, which is made to solve theseproblems, is to provide a spectacle lens with high performance whichimproves visual acuity and to provide a designing method of the same.

DISCLOSURE OF THE INVENTION

[0022] In order to solve the above-described problems, a first inventionis

[0023] a spectacle lens designing method in which an eyeball motion(Listing's Law) is taken into consideration, and which is characterizedin that a merit function used in optimization calculation processing oflens designing includes a visual acuity evaluation function (log MAR)derived from a visual acuity measured value V,

[0024] where the visual acuity evaluation function (log MAR) isexpressed by the following equation (1), letting a curvature of field bean aberration of a spectacle lens and a residual astigmatism be anastigmatism defined from spectacle lens designing in which the Listing'sLaw is taken into consideration:

log MAR=log₁₀(1/V(curvature of field, residual astigmatism))   (1)

[0025] A second invention is a spectacle lens designing method which ischaracterized in that, in the spectacle lens designing method of thefirst invention, letting the visual acuity measured value V beV=2^(−x·K) (where K={(residual S diopter+residual Cdiopter/2)²+(residual C diopter/2)²}^(1/2) and X is a coefficientbetween 0.5 and 2 according to actual measurement data), the visualacuity evaluation function (log MAR) is expressed by the followingequation (2) which is an approximate equation:

log MAR=X×log₁₀2×{curvature of field²+(residual astigmatism/2)²}^(1/2)  (2)

[0026] A third invention is a spectacle lens designing method which ischaracterized in that, in the spectacle lens designing method of thefirst invention, the merit function includes an evaluation function on adistortion aberration (residual distortion aberration DIST) and theevaluation function is expressed by the following equation (3):

residual distortion aberration DIST=Sign×100×(absolute value of residualvisual angle magnification/absolute value of central visual anglemagnification M ⁰ )   (3)

[0027] where:

[0028] the residual visual angle magnification is the distortionaberration defined from the spectacle lens designing in which theListing's Law is taken into consideration; and

[0029] Sign is a positive/negative sign.

[0030] A fourth invention is a spectacle lens designing method which ischaracterized in that, in the lens designing method according to any oneof the first invention to the third invention, the merit function isused in optimization calculation of lens designing of a bi-asphericallens in which a front surface is an axially symmetrical asphericalsurface and a rear surface is an aspherical surface expressed by thefollowing equation (4): $\begin{matrix}{{Z2} = {{{c(\theta)} \cdot {r^{2}/\left( {1 + \sqrt{1 - {\left( {1 + {k(\theta)}} \right) \cdot {c(\theta)}^{2} \cdot r^{2}}}} \right)}} + {\sum\limits_{n}{{a\left( {n,\theta} \right)} \cdot r^{n}}}}} & (4)\end{matrix}$

[0031] where:

[0032] c(θ), k(θ) are functions for an azimuth θ;

[0033] a(n, θ) is a function for an n degree of a distance r and theazimuth θ;

[0034] as for a definition domain of the azimuth θ, 0 degree to 90degrees represents 0 degree to 360 degrees due to plane symmetry of anastigmatic lens;

[0035] c(θ) is a curvature of a lens center and is expressed by thefollowing equation (5) based on the Euler's theorem, letting a curvatureof a spectacle principal meridian in the Gaussian curve theorem be c(0)at 0 degree and c(90) at 90 degrees. In this case, 0 degree is aspherical diopter axis and 90 degrees is a cylinder diopter axis;

c(θ)=c(0)·cos² θ+c(90)·sin² θ  (5)

[0036] k(θ), which is similar to c(θ) above, represents an equation inwhich the sign c is replaced by the sign k in the above equation (5);and

[0037] a(n, θ) satisfies requirements of plane continuity and planesymmetry, is a surface further satisfying a requirement of a surfacewhich is capable of controlling an aberration due to an angle deviationwhich occurs due to the Listing's Law, and further satisfies thefollowing conditions {circle over (1)} to {circle over (4)}:

[0038]{circle over (1)}: having a functional relation of the azimuth θfrom 0 degree to 90 degrees;

[0039]{circle over (2)}: a linear differential coefficient of theazimuth θ is 0 from 0 degree to 90 degrees;

[0040]{circle over (3)}: a high degree differential coefficient iscontinuous; and

[0041]{circle over (4)}: having a control parameter group Ps(n) which iscapable of controlling a value of a(n, θ) at an angle θ of a functionbetween the azimuths 0 degree and 90 degrees (where 1 to 3 arepreferable for s, and n signifies a degree in the above equation (4)).

[0042] A fifth invention is a spectacle lens designing method which ischaracterized in that, in the spectacle lens designing method accordingto the fourth invention, a(n, θ) in the above equation (4) is expressedby the following equation (6) which is a quartic polynominal of theazimuth θ, letting a be a(n, 0), a(n, 45), and a(n, 90) when the azimuthθ is 0 degree, 45 degrees, and 90 degrees respectively:

a(n, θ)=a(n, 0)+(−11·a(n, 0)+16·a(n, 45)−5·a(n, 90))·θ²/(4·90²)+(9·a(n,0)−16·a(n, 45)+7·n(n, 90))·θ³/(4·90³)+(−2·a(n, 0)+4·a(n, 45)−2·a(n,90)·θ⁴/(4·90⁴)   (6)

[0043] where a control parameter is one for the degree n of the distancer from the center and a control parameter P1(n) is a(n, 45).

[0044] A sixth invention is a spectacle lens designing method which ischaracterized in that, in the spectacle lens designing method accordingto the fourth invention, a(n, θ) in the above equation (4) is expressedby the following equation (7), letting a be a(n, 0) and a(n, 90) whenthe azimuth θ is 0 degree and 90 degrees respectively:

a(n, θ)=a(n, 0)·cos² θ+a(n, 90)·sin² θ+P1(n)·sin²(2·θ)   (7)

[0045] where a control parameter is one control parameter P1(n) for thedegree n of the distance r from the center.

[0046] A seventh invention is a spectacle lens which is characterized inthat it is designed by the spectacle lens designing method according toany one of the first invention to the sixth invention.

BRIEF DESCRIPTION OF DRAWINGS

[0047]FIG. 1 is an explanatory view of a spectacle lens designing methodaccording to an embodiment of the present invention;

[0048]FIG. 2 is an explanatory view of an extended DIST;

[0049]FIG. 3 is a view showing Table 1 in which lens data in Example 1are listed;

[0050]FIG. 4 is a view showing Table 2 in which lens data in Comparisonexample of Example 1 are listed;

[0051]FIG. 5 is a view showing log MAR in Example 1;

[0052]FIG. 6 is a view showing log MAR in Comparison example of Example1;

[0053]FIG. 7 is a view showing Table 3 in which lens data in Example 2are listed;

[0054]FIG. 8 is a view showing the distribution of a first quadrant ofthe extended DIST in Example 2;

[0055]FIG. 9 is a view showing the distribution of the extended DIST inComparison example of Example 1;

[0056]FIG. 10 is a view showing log MAR in Example 2;

[0057]FIG. 11 is an explanatory view of a spectacle lens designingmethod in a prior art; and

[0058]FIG. 12 is a view showing actual measurement values of visualacuity.

[0059] VS . . . rear vertex spherical surface; V . . . rear vertex; W .. . reference point of focal length; R . . . center of rotation ofeyeball; FPS . . . far point sphere; Ft . . . focus in radial tangentdirection; Fs . . . focus in sagittal direction; D . . . image on farpoint sphere; Ws . . . reference point of focal length of ray passing onS axis; Wc . . . reference point of focal length of ray passing on Caxis; FPS: far point sphere in S axis direction; FPC: far point spherein C axis direction; Fst: focus in S axis direction of ray passing on Saxis; Fss . . . focus in C axis direction of ray passing on S axis; Fct. . . focus in S axis direction of ray passing on C axis; Fcs . . .focus in C axis direction of ray passing on C axis; DS . . . image onfar point sphere in S axis direction; DC . . . image on far point spherein C axis direction; P . . . visual angle magnification evaluationpoint; MO . . . reference visual angle magnification in P direction; M .. . visual angle magnification at position P

BEST MODE FOR CARRYING OUT THE INVENTION

[0060] As a paper on retina and brain processing regarding visualacuity, Optmetric Monthly, November: 31-32 1981: written by Robert N.Kleinstein (hereinafter, referred to as Paper 1) is available.

[0061] A drawing in the above Paper 1 shows a view in which a visualacuity measured value is expressed by a fraction visual acuity value,taking S diopter and C diopter as spectacle terms in a horizontal axisand a vertical axis respectively and an experiment of measuring visualacuity of a spectacle wearer with his/her spectacles taken off isconducted. In order to use this Paper 1 as an evaluation function of amerit function in spectacle lens designing, the measured values aremodified in such a manner that the signs of the horizontal axis value Sand the vertical axis value C are reversed, namely, the residual Sdiopter and the residual C diopter are taken in the horizontal axis andthe vertical axis respectively to obtain evaluation data showing how thevisual acuity decreases when a subject person having normal visualacuity wears spectacles with an aberration, reversely to the aboveexperiment.

[0062] In FIG. 12 described above, data for the age of 5 to 15, 25 to35, and 45 to 55 are provided as actual measurement data, but since itis preferable to use a virtual visual acuity measured value not affectedby an adjusting power, the data for the age of 45 to 55 were used fromPaper 1 for convenience sake.

[0063] The residual S diopter and the residual C diopter mentioned aboveare correlated to an astigmatism and a curvature of field derived fromoptical calculation as described later. In the spectacle lens designingin the prior art in which the Listing's Law is not taken intoconsideration, however, the astigmatism and the curvature of fieldcannot be calculated accurately in regions in which an eyeball does notrotate along two spectacle principal meridians as previously described.Therefore, a spectacle lens designing system in which the Listing's Lawis taken into consideration and which includes new lens aberration(astigmatism and curvature of field) calculation is required in order touse the measured values of the visual acuity measurement in Paper 1mentioned above as an evaluation function on the entire surface of alens.

[0064] (Spectacle Lens Designing System Including Lens Aberration(Astigmatism and Curvature of Field) Calculation)

[0065]FIG. 1 is a view explaining one model to be a factor in aspectacle lens designing method according to an embodiment of thepresent invention, and FIG. 11 is a view explaining a model in a priorart with which the above model is compared.

[0066] In a case of rays passing S and C axes of an astigmatic lensshown in FIG. 1, calculation similar to the case shown in FIG. 11 of adesigning system in the prior art is valid.

[0067] However, on an axis in a lens radiation direction other than theS and C axes of the astigmatic lens in FIG. 1, it is necessary tocalculate the astigmatism and the curvature of field with an eyeballmotion taken into consideration, which are calculated by the followingmethod.

[0068] Hereinafter, the correlation of the residual S diopter and theresidual C diopter with the astigmatism and the curvature of field inthe spectacle lens designing system in which the Listing's Law is takeninto consideration will be simply explained.

[0069] I. (Astigmatism and Curvature of Field)

[0070] In Prior art 1 in which the Listing's Law is taken intoconsideration, when the rotation is in a different direction from thespectacle principal meridians, the angle between the spectacle principalmeridians and coordinate axes rotating according to the Listing's Lawdoes not become 0. When the angle deviation as described in the abovePrior art 1 occurs, the astigmatism, even when, typically, it is anastigmatism having an absolute value of the astigmatism equal to anabsolute value of a reference astigmatism (an astigmatic amount and acylinder axis at the center of a lens), has a direction like a vectorvalue so that a residual astigmatism whose value is not 0 newly occurs.

[0071] As for a calculation method of the above residual astigmatism,methods of calculating an astigmatic lens and of the residualastigmatism of the astigmatic lens as disclosed in, for example, Priorart 1 are applicable.

[0072] Meanwhile, the curvature of field as another factor does notchange due to the coordinate change according to the Listing's Law sincethe curvature of field is a scalar amount not related to a vector.

[0073] I-1. (Residual Astigmatism)

[0074] Therefore, the correlation of the aforesaid residual astigmatismand curvature of field with the residual S diopter and the residual Cdiopter is as follows:

[0075] (1) When the residual astigmatism is positive, their correlationis expressed by the following equations (a), (b):

residual S diopter=curvature of field+residual astigmatism/2   (a)

residual C diopter=residual astigmatism   (b)

[0076] (2) When the residual astigmatism becomes negative in opticalcalculation, their correlation is expressed by the following equations(c), (d) based on an idea similar to diopter conversion of spectaclessince the residual C diopter is defined as positive:

residual S diopter=curvature of field−residual astigmatism/2   (c)

residual C diopter=−residual astigmatism   (d)

[0077] II. (Deriving Merit Function in Which Nonlinear Nature of LivingHuman Body in View of Optical Performance is Taken into Consideration)

[0078] On analyzing FIG. 12 in the aforesaid Paper 1, it is first foundout that the horizontal axis (residual S diopter) is not symmetricalwith respect to the origin. Furthermore, the vertical axis (residual Cdiopter) has also nonlinear data peculiar to the living human body.

[0079] For example, when visual acuity values with the same absolutevalue on the horizontal axis and with different signs are examined, itis clear that the functional relation is not simple. Therefore, whenoptimization calculation is directly done in the optical calculationwithout taking the nonlinear nature peculiar to the living human bodyinto consideration, this does not always indicate that visual acuitythrough a designed lens is improved since the visual acuity value isnonlinear relative to an optical performance value.

[0080] Therefore, in the embodiment of the present invention, aninterpolation function V is first prepared from the data on the visualacuity measured values in FIG. 12. Concretely, an equation (e) by whichthe interpolation function V can be calculated even with continuousresidual S diopter and residual C diopter is prepared using a generallyknown interpolation method, taking the visual acuity values forhorizontal axis values (residual S diopter) and vertical axis values(residual C diopter) by discrete values (every 0.1 to 1 diopter).

[0081] This is expressed by the following equation:

interpolation function V=V(residual S diopter, residual C diopter)   (e)

[0082] Using this interpolation function V, the aforesaid residualastigmatism and curvature of field of the lens are calculated, and theyare substituted for the residual S diopter and the residual C diopter inthe equations (a), (b) or the equations (c), (d).

[0083] Then, the optical value and the visual acuity value arecorrelated in such a manner that a right side is obtained by the opticalcalculation and a left side is the visual acuity value by actualmeasurement as in the following equation (f):

interpolation function V=V(curvature of field, residual astigmatism)  (f)

[0084] The equation (f) in this state can be used as an evaluationfunction, but since nonlinearity is high, it is not the best state forthe optimization calculation.

[0085] Therefore, it is further transformed to the following equation(g) expressed by a visual acuity evaluation function log MAR, which is adefinition equation for representing visual acuity.

log MAR=log₁₀(1/V(curvature of field, residual astigmatism))   (g)

[0086] Through the above processes, the evaluation function in which thenonlinear nature of the living human body from the optical performancepoint of view is taken into consideration is derived.

[0087] The visual acuity of the living human body of course changes to agreat extent depending on age, a measurement environment, and so on.

[0088] In actual application, however, the above-described basic methodrequires a large calculation amount in the optimization calculation.

[0089] Therefore, instead of the equation (e) by which the aforesaidinterpolation function V can be calculated, simple approximate equationssuch as the following equations (h), (i) can be used:

V=2^(−X·K)   (h)

[0090] where,

[0091] K is expressed by the following equation (i):

K={(residual S diopter+residual C diopter/2)²+(residual Cdiopter/2)²}^(1/2)   (i)

[0092] X is a coefficient between 0.5 and 2 according to actuallymeasured data.

[0093] In the above case, V may be used as the evaluation function as itis, but the correlation with the visual acuity evaluation function logMAR is expressed by the following equation, as explained in theaforesaid basic method.

log MAR=X×log₁₀2×{(curvature of field²+(residual astigmatism/2)²)^(1/2)  (i)

[0094] Furthermore, the approximate equations can be transformed byincluding measured values according to age besides data in the materialfor actually measured visual acuity and by using other visual acuitymeasurement data. For example, the transformation of the equation (h)such as the following equation V=3^(−K) is possible under the conditionof within a variable range of X. In this case, the equation (j) becomesas follows:

log MAR=log₁₀3×{(curvature of field²+(residual astigmatism/2)²)^(1/2)

[0095] III. (Distortion Aberration with the Listing's Law Taken intoConsideration)

[0096] Furthermore, as an aberration to be corrected for spectacles,which is not related to a visual acuity value, there is a distortionaberration.

[0097] This is widely known as a cause of sway and distortion occurringmainly at the beginning when one starts to wear spectacles.Conventionally, the distortion of spectacles is expressed as a visualangle magnification (for example, refer to KOHGAKU (OPTICS), Vol. 19,No. 10 “Futatabi Kakubairitsu nitsuite (On Angle Magnification Again)”written by Kazuo Miyake, and so on).

[0098] When this is expressed by an equation, letting a central visualangle magnification be M₀, the following equation (k) is obtained:

central visual angle magnification M ₀=1 im _(exit angle→0)(tan(exitangle)/tan(incident angle))   (k)

[0099] Here, the central visual angle magnification M₀ can be easilycalculated by paraxial optical calculation. The central visual anglemagnification M₀ will be simply explained. When an emergent ray passesthe center of eyeball entrance pupil, the central visual anglemagnification M₀ is called a spectacle magnification.

[0100] Further, letting a peripheral visual angle magnification be M,this visual angle magnification M is expressed by the following equation(l):

peripheral visual angle magnification M=tan(exit angle)/tan(incidentangle)   (1)

[0101] Then, the distortion aberration (DIST) of the spectacles isexpressed by the following equation (m) based on the equations (k), (l):

distortion aberration DIST=100×((M/M ₀)−1)   (m)

[0102] Incidentally, in the model in FIG. 1, the emergent ray passes thecenter of rotation of the eyeball and the distortion aberration DIST iscalled a dynamic distortion aberration of the spectacles.

[0103] Here, on studying the equation (m), a residual distortionaberration DIST occurs due to the difference (angle deviation) of anaxis direction since the distortion aberration DIST, even when it is theaberration DIST with the same amount, is a vector value, similarly tothe previous explanation on the astigmatism.

[0104] Therefore, the central visual angle magnification M₀ and theperipheral visual angle magnification M in the prior art are calculatedas the distortion aberration DIST when they are in the same direction.

[0105] For example, if the central visual angle magnification M₀ and theperipheral visual angle magnification M in the same direction are thesame amount, the distortion aberration DIST is calculated as thedistortion aberration DIST=0 by the equation (m).

[0106] Since the aforesaid angle deviation caused by the eyeball motionis included in the calculation, the central visual angle magnificationM₀ and the peripheral visual angle magnification M are both extendedlydefined as vector amounts.

[0107] Then, when the lens is an astigmatic lens, the rotational visualangle magnification M₀ becomes a vector value having a different valuein the radiation direction at a lens diopter reference point (usually,the center part of the lens).

[0108] When a residual visual angle magnification is defined as a valueobtained by subtracting the central visual angle magnification from theperipheral visual angle magnification M, this residual visual anglemagnification is expressed by the following equation:

residual visual angle magnification=peripheral visual anglemagnification M−central visual angle magnification M ₀

[0109] The extended definition of the distortion aberration of thespectacles according to the embodiment of the present invention in whichthe Listing's Law is taken into consideration becomes the followingequations (n), (o):

residual visual angle magnification=peripheral visual anglemagnification M−central visual angle magnification M ₀   (n)

residual distortion aberration DIST=Sign×100×(absolute value of residualvisual angle magnification/absolute value of central visual anglemagnification M ₀)   (o)

[0110] where Sign is defined as a positive/negative sign of a scalarproduct of the residual visual angle magnification and the centralvisual angle magnification M₀.

[0111]FIG. 2 is a view showing the correlation of the equations (n) and(o).

[0112] Through the above, a residual distortion aberration equation ofthe spectacles in which the Listing's Law is taken into consideration isderived and it is further incorporated in the merit function.

[0113] IV. (Preparation of Merit Function)

[0114] In the spectacle lens designing method according to theembodiment of the present invention, the state in which a ray passes alens is assumed and simulation calculation is done by the ray tracingmethod, and usually, about 5 to about 10 axially symmetrical lenses canbe adopted and about 15 to about 10000 lenses according to thisembodiment can be adopted to calculate the aforesaid equations (g), (o).

[0115] In the case of the aforesaid equation (g), different values areobtained depending on the evaluated object distance. Determination onwhich object distance is to be taken is made in consideration of a lenscharacteristic and so on.

[0116] For example, strictly speaking, there is no actually measuredvisual acuity value of near vision in an equation (p) described later,but responses to the residual S diopter and the residual C diopter canbe calculated assuming that they are similar to those in a case of farvision.

[0117] Furthermore, it is said that the dynamic distortion aberration ofthe spectacles is not related to the object distance theoretically, butactually, no clear material exists on how to deal with the distributionof the visual acuity and the distortion, and so on. Therefore, they canbe freely set within a range not departing from the object of thedesigning.

[0118] From the above, the merit function according to the presentinvention, which is a combined function of evaluation functions and is asingle evaluation criterion, becomes the following equation (p).$\begin{matrix}{{{{merit}\quad {function}} = {{a \times {\sum\limits_{n}\left( {{u_{n} \cdot {far}}\quad {vision}\quad \log \quad {MAR}_{n}} \right)^{2}}} + {b \times {\sum\limits_{n}\left( {{v_{n} \cdot {near}}\quad {vision}\quad \log \quad {MAR}_{n}} \right)^{2}}} + {c \times {\sum\limits_{n}\left( {{w_{n} \cdot {residual}}\quad {DIST}_{n}} \right)}}}}\quad} & (p)\end{matrix}$

[0119] Here, a, b, c are weight distribution of respective evaluationfunctions; u, v, w are weight distribution at respective evaluationpoints; and n is a lens evaluation point. Of course, the idea (=notadopted) that the weight distribution is 0 (zero) is included, butnaturally, they never become 0 synchronously.

[0120] However, few objective experimental data which determines theweight is available, and in actual application, the weight distributionis carried out in consideration of the object of using the lens, andaesthetic, economical, optical factors and so on.

[0121] Moreover, it is possible to add to the merit function of thepresent invention items not directly related to the visual acuity suchas a lens form and so on.

[0122] The aforesaid merit function (p) is made optimized using theoptimization method. This optimization method is as explained in thesection of the background art previously described (for example, theaforesaid Japanese Patent Publication No. Hei 2-38930 and so on).

[0123] The aforesaid merit function (p) will be studied from theviewpoint of the degree of freedom of designing a lens refractivesurface.

[0124] When a front surface and a rear surface of the lens are freecurved surfaces which can be transformed freely under the restrictivecondition that the diopter of the lens is fixed based on a prescriptionvalue, a first term or a second term in the merit function can be madezero by the transformation of these two surfaces.

[0125] Specifically, at a certain object distance, the astigmatism andthe curvature of field which are constituent factors of the visualacuity evaluation function log MAR can both be made 0.

[0126] However, when an aesthetic factor of its appearance is added andan economical viewpoint such as manufacturing cost is taken intoconsideration in designing the front surface which is a surface on anobject side of the lens, for example, when the restrictive condition ofan axially symmetrical aspherical surface is added, it is difficult tosynchronously make the residual astigmatism and the curvature of field 0on the entire surface of the spectacle lens at a certain objectdistance.

[0127] Still more, it is generally difficult to make the residualdistortion aberration DIST 0 in the surface structure where lens diopterexists, without influencing other evaluation functions. Therefore, acoefficient and weighting are dealt as design items. Furthermore, fromthe viewpoint of the degree of freedom of designing, when the structureof the front surface is fixed, for example, by the condition of a sphereand so on, the degree of freedom of designing is restricted, and itbecomes difficult to control a third term in the merit function, namely,the residual distortion aberration DIST.

[0128] In other words, the merit function is a function in which theaberrations are complicatedly combined as describe above, and if thesurface has a restriction such as a sphere when the merit function isoptimized by the optimization, the optimization is influenced by therestriction.

[0129] Therefore, it is preferable that the front surface and the rearsurface of the spectacle lens are both set in such a manner that theycan be designed by free transformation, thereby enabling the meritfunction to be freely controlled and increasing the degree of freedom ofdesigning.

[0130] V. (Design of Bi-Aspherical Lens)

[0131] Here, as a design example in which the degree of freedom ofdesigning is taken into consideration, the explanation will be given ona spectacle lens consisting of aspherical surfaces on both sides, whichenables the above merit function to be optimized by the optimizationcalculation with high precision and with high calculation efficiency.

[0132] Since according to the Listing's Law, the rotation is made in aradiation direction from the first eye position of the eye as ispreviously described, a corresponding expression of a lens surfacebecomes directly corresponding to the eyeball motion when it isexpressed by a spherical coordinate system and a cylindrical coordinatesystem with the lens center being the origin.

[0133] However, when it is expressed by other coordinate systems, forexample, an orthogonal coordinate system and so on, a high degreecoefficient becomes necessary, though they are mathematicallyequivalent, in order to bring about an equivalent effect in numericalcalculation, and consequently, a calculation error is increased.

[0134] Furthermore, though the aforesaid spline curved surface, a NURBScurved surface, and so on are also capable of expressing very freecurved surfaces, they are basically the orthogonal coordinate systemsimilarly to the above so that the similar problem occurs in thenumerical calculation.

[0135] Therefore, in this embodiment, an aspherical surface equation ofthe cylindrical coordinate system is used as a preferable method (referto, for example, Prior art 2 for the aspherical surface equation of thecylindrical coordinate system in detail).

[0136] (Aspherical Surface Equation Expressing Refractive Surface Shapeof Front Surface)

[0137] A lens height Z1 of the front surface, which is expressed by thefollowing equation (q), is expressed as an equation of a lens crosssection. $\begin{matrix}{{Z1} = {c \cdot {r^{2}/\left( {1 + \sqrt{1 - {\left( {1 + k} \right) \cdot c^{2} \cdot r^{2}}} + {\sum\limits_{n}{{a(n)} \cdot r^{n}}}} \right.}}} & (q)\end{matrix}$

[0138] In the first term of the right side, which is a rotationalquadric surface; c is a center curvature; k is a conic coefficient; andr is a distance between the position of the lens projected on ahorizontal plane of the cylindrical coordinate system and the origin,and in the second term, which is a deviation from the rotational quadricsurface, n, though it takes values from 2, usually takes values from 4to 12 since it interferes with the first term. a(n) is an n degreecoefficient of r and is an amount called an aspherical coefficient.

[0139] V-1 (Aspherical Surface Equation Expressing Refractive SurfaceShape of Rear Surface)

[0140] An equation of the rear surface of the present invention is thefollowing equation (r): $\begin{matrix}{{Z2} = {{c(\theta)} \cdot {r^{2}/\left( {1 + \sqrt{\left. {1 - {\left( {1 + {k(\theta)}} \right) \cdot {c(\theta)}^{2} \cdot r^{2}}} \right)} + {\sum\limits_{n}{{a\left( {n,\theta} \right)} \cdot r^{n}}}} \right.}}} & (r)\end{matrix}$

[0141] Here, c(θ), k(θ) are functions for an azimuth θ. a(n, θ) is afunction for the n degree of the distance r and the azimuth θ. Due toplane symmetry of the astigmatic lens, as for a definition domain of theazimuth θ, 0 degree to 90 degrees can represent 0 degree to 360 degrees.Here, c(θ) is a curvature of the lens center, and letting the curvatureof the two principal meridians orthogonal to each other be c(0) andc(90) at 0 degree and 90 degrees respectively, as is stated in theGaussian curve theorem, the following equation (s) is obtained from theEuler's theorem.

[0142] In the case of the lens, 0 degree and 90 degrees are taken in thespherical diopter axis and in the astigmatic diopter axis respectively,and c(θ) is expressed by the following equation (s):

c(θ)=c(0)·cos² θ+c(90)·sin² θ  (s)

[0143] k(θ) is similar to the above equation (s) and becomes an equationin which the sign c in c(θ) is replaced by the sign k.

[0144] a(n, θ) satisfies requirements of plane continuity and planesymmetry, is a surface further satisfying a requirement of a surfacewhich is capable of controlling an aberration due to an angle deviationwhich occurs due to the Listing's Law, and satisfies the followingconditions {circle over (1)} to {circle over (4)}:

[0145]{circle over (1)}: having a functional relation of the azimuth θfrom 0 degree to 90 degrees;

[0146]{circle over (2)}: a linear differential coefficient of theazimuth θ is 0 from 0 degree to 90 degrees;

[0147]{circle over (3)}: a high degree differential coefficient iscontinuous; and

[0148]{circle over (4)}: having a parameter group: Ps(n) which iscapable of controlling a value a(n, θ) at an angle θ of a functionbetween the azimuths 0 degree and 90 degrees (where 1 to 3 arepreferable for the number of s from the viewpoint of calculation speedand calculation efficiency, and n signifies a degree in the aboveequation (r)).

[0149] Concretely, for example, (in a case when the functional relationis a polynominal of an angle) letting the polynominal be a quarticpolynominal of the azimuth θ, and a at 0 degree, 45 degrees, 90 degreesbe a(n, 0), a(n, 45), a(n, 90) respectively, a(n, θ) becomes thefollowing equation (t):

a(n, θ)=a(n, 0)+(−11·a(n, 0)+16·a(n, 45)−5·a(n, 90))·θ²/(4·90²)+(9·a(n,0)−16·a(n, 45)+7·a(n, 90))·θ³/(4·90³)+(−2·a(n, 0)+4·a(n, 45)−2·a(n,90))·θ⁴/(4·90⁴)   (t)

[0150] In this case, the above-mentioned control parameter in {circleover (4)} is one for the degree n of the distance r from the center andthe control parameter P1(n) is a(n, 45).

[0151] (in a case when the functional relation is not a polynominal ofan angle, for example, is a trigonometric function)

[0152] a(n, θ) is expressed by the following equation (u), letting a bea(n, 0) and a(n, 90) when the azimuth θ is 0 degree and 90 degreesrespectively in the following function and letting the control parameterwhich is one for the degree n of the distance r from the center be P1(n)similarly to the above:

a(n, θ)=a(n, 0)·cos² θ+a(n, 90)·sin² θ+P(1, n)·sin²(2·θ)   (u)

[0153] The equations (t), (u) both satisfy the above conditions {circleover (1)} to {circle over (4)}.

[0154] Thus, there exist various equations satisfying the aboveconditions {circle over (1)} to {circle over (4)}.

EXAMPLE 1

[0155] In Example 1, a spectacle lens is designed using the evaluationfunction on visual acuity of the present invention, and the outline ofthe designing procedure thereof will be explained below.

[0156] (Step 1): To Set a Basic Design Lens Form of Front and RearRefractive Surfaces

[0157] In this example, a bi-aspherical lens form which has the highestdegree of freedom of designing is selected, with the front surface beingan aspherical surface which is axially symmetrical and expressed by theabove equation (q) and with the rear surface being an aspherical surfaceexpressed by the above equation (r).

[0158] (Step 2): To Set Fixed Conditions and Variable Conditions of aShape Determining Factor Parameter

[0159] The design conditions are, in the prescription values, aspherical diopter is −7.00 D, a cylindrical diopter is −2.00 D, arefractive index (ne) is 1.7, a lens diameter is 75 mm, and a lenscenter thickness is 1 mm, as shown in FIG. 7.

[0160] In the above aspherical surface equations (q) and (r), k(θ) is 0and the equation (t) is applied to a(n, θ).

[0161] Note that coefficients in the equations are as shown in FIG. 7.

[0162] (Step 3): To Set the Merit Function and a Target Value of theOptimization Calculation

[0163] The above equation (p) is used for the merit function and itscondition is a=1, b=0, c=0, and u=1.

[0164] The equation (j) is used for the equation of the visual acuityevaluation function log MAR and its condition is X=2.

[0165] (Step 4): Optimization Calculation

[0166] Based on set lens evaluation points, their evaluation is madeusing the aforesaid merit function by the ray tracing method, opticalperformance is evaluated, simulation calculation is repeated byoperating transformation parameters constituting the lens refractivesurface until the predetermined target value is obtained, and theoptimization calculation is carried out.

[0167] At this time, an optimal solution is calculated under thecondition that the curvature of the front surface does not becomenegative (incidentally, a lens whose curvature of the front surfacebecomes negative is described in Prior art 1, but it cannot be said tobe aesthetically optimal since a reflected light is strong).

[0168] In this example, the final refractive surface shape is determinedby fixing the design condition that the front surface is aspherical andby varying the shape of the rear surface so as to satisfy theprescription values. Obtained lens data (final lens performance dataafter the optimization is finished) are shown in Table 1 in FIG. 3.

[0169] Further, the distribution of the log MAR visual acuity values inExample 1 in the case of the lens data in FIG. 3 is shown in FIG. 5.

[0170] 64% of a thin portion in the lens center part produces preferablevisual acuity whose log MAR visual acuity value is 0.2 or lower.

[0171] A Percival Form lens in which the curvature of field is reducedunder the same condition as that of Example 1 is shown for comparison.

[0172] Obtained lens data and the distribution of the log MAR visualacuity values are shown in Table 2 in FIG. 4 and FIG. 6 respectively.

[0173] The curvature of field of this lens is preferable, but 56% of thethin portion of the lens center part produces the preferable visualacuity whose log MAR visual acuity value is 0.2 or lower.

[0174] Thus, it is clear that the preferable visual acuity range isobtained in FIG. 5, compared with that in FIG. 6, and the control of theevaluation function of visual acuity can sufficiently be performed sothat the expected effect is obtained.

EXAMPLE 2

[0175] In Example 2, an evaluation function on the residual distortionaberration DIST is further added to Example 1 to design a spectaclelens. Since the lens does not produce visual acuity and the optimalsolution cannot be obtained when only the residual distortion aberrationDIST is used in the aforesaid merit function equation (p), the log MARvisual acuity value and the residual distortion aberration DIST arebalanced in the equation (p).

[0176] In the equation (p), a=1, b=0, c=0.02, u=1, and w=1, and theequation (j) is used for the equation for the visual acuity evaluationfunction log MAR.

[0177] The equations (q), (r) are used for the bi-aspherical surfaceequation, k(θ) is 0, and the equation (t) is applied for a(n, θ).

[0178] The data in FIG. 5 in Example 1 are used for the front surface.Though this is not a suitable condition for greatly improving theresidual distortion aberration DIST since a fixed condition is set forthe front and rear surfaces, the optimization calculation is done underthe above condition since it is indicated that the residual distortionaberration DIST can be controlled within a certain range. Obtained lensdata are shown in Table 3 in FIG. 7.

[0179]FIG. 8 is a table showing the distribution of the residualdistortion aberration DIST in the first quadrant. The lowest right endis the lens center, where the residual distortion aberration DIST is 0.The horizontal axis is a lens exit angle in the lens S axis direction,which is shown for every 3 degree pitch, and similarly, the verticalaxis is the same in the lens C axis direction.

[0180]FIG. 9 shows, as a comparison example, the distribution of theresidual distortion aberration DIST under the condition in FIG. 5 inExample 1 in which the residual distortion aberration DIST is notevaluated as an evaluation function. The final values of the horizontalaxis and the vertical axis in FIG. 8 are 43% and 60%, and the finalvalues of the horizontal axis and the vertical axis in FIG. 9 are 44%and 63%. Since a smaller value signifies more preferable state in thiscase, it is clear that the control of the evaluation function for theresidual distortion aberration DIST can be sufficiently performed andthe expected effect is obtained.

[0181] Incidentally, a distribution view of the log MAR visual acuityunder the condition in FIG. 7 is shown in FIG. 10. The range where thelog MAR visual acuity value is 0.2 or lower is 53%, and in improving theresidual distortion aberration DIST and the log MAR visual acuity value,they are in a trade-off correlation in which, when one value isimproved, the other value is lowered.

[0182] However, since sway is usually sensed at a peripheral portion, itis also possible to improve the residual distortion aberration DIST insuch a manner that the distribution of the weights (u, v, w) at therespective evaluation points in the aforesaid merit function equation(q) is devised so as to give a higher weight to the log MAR visualacuity value in the center portion and to sacrifice the log MAR visualacuity value in the peripheral portion.

[0183] The merit function including the visual acuity evaluationfunction according to the present invention is used for thebi-aspherical type lens having a single focus in this example. However,since the technical structure of the invention is characterized in thatthe visual acuity evaluation function is used as the evaluation functionof the merit function used in the optimization calculation, it is notlimited by the refractive shape of the lens surface, and can be used indesigning of all lenses including progressive refracting surfaces.

[0184] For example, in a progressive-power lens, other factors such as adistance portion, a near portion, a progressive zone are added besidesweighting on the lens central portion and peripheral portion, which isused in a case of ordinary lenses, and near vision weighted design, farvision weighted design, intermediate vision weighted design, and so onare also added to the object of the designing. However, since theprogressive-power lens uses the aspherical lens surface similarly tothis example when classified in terms of a lens surface, the presentinvention is applicable to the progressive-power lens by making themerit function according to the present invention correspond to theobject of its designing, appropriately setting the weight distributionat the evaluation points, setting target diopter and a target distortionaberration, and changing these design items.

[0185] The present invention is especially useful for the designing inwhich the Listing's Law is taken into consideration since an accuratesimulation can be carried out.

[0186] Furthermore, the same thing can be said for a lens whose rearsurface is a fusion surface of an aspherical surface and an astigmaticsurface.

[0187] In this example, data in Optmetric Monthly, November: 31-32 1981:written by Robert N. Kleinstein are used as a paper on the processing inthe retina and the brain regarding visual acuity. The present invention,however, is not limited to this, and any data can be used and the visualacuity evaluation function included in the present invention can bederived from the data, as long as they are data on the visual acuitymeasured value in which, for example, visual acuity and diopter arecorrelated.

[0188] Furthermore, in a manufacturing method, in the case of, forexample, the bi-aspherical lens in this example, the front surface ismade to be an axially symmetrical aspherical surface and the rearsurface is made to be an aspherical lens of the free curved surface, sothat a semi-finished lens can be used, which is effective in terms oftime and cost. In other words, when a plurality of axially symmetricalaspherical lens having a predetermined common base curve are prepared inadvance as described above, the semi-finished lens is first selectedaccording to the prescription after receipt of order, and thereafter,its rear surface is designed, it is more advantageous than to design aconvex surface and a concave surface after each receipt of order andprepare a finished lens.

[0189] Moreover, by the aforesaid fixing of the design, it becomespossible to prepare finished products in advance in stock according tothe prescription.

[0190] Industrial Availability

[0191] As detailed above, in contrast to a spectacle lens designing in aprior art in which the performance of a spectacle lens is evaluated onlywith indexes such as an optical amount on the retina and aberrationsbased on the technical idea that the performance of the spectacle lensis improved as optical performance is made higher, it becomes possibleto design a spectacle lens based on a simulation on a living human body,in which the viewpoints of the processing in the retina and the brainand of an eyeball motion are taken into consideration, and a spectaclelens with higher performance can be obtained.

1. A spectacle lens designing method in which an eyeball motion(Listing's Law) is taken into consideration, wherein a merit functionused in optimization calculation processing of lens designing includes avisual acuity evaluation function (log MAR) derived from a visual acuitymeasured value V, where the visual acuity evaluation function (log MAR)is expressed by the following equation (1), letting a curvature of fieldbe an aberration of a spectacle lens and a residual astigmatism be anastigmatism defined from spectacle lens designing in which the Listing'sLaw is taken into consideration: log MAR=log₁₀(1/V(curvature of field,residual astigmatism))   (1)
 2. A spectacle lens designing methodaccording to claim 1, wherein, letting said visual acuity measured valueV be V=2^(−x·K) (where K={(residual S diopter+residual Cdiopter/2)²+(residual C diopter/2)²}^(1/2) and X is a coefficientbetween 0.5 and 2 according to actual measurement data), said visualacuity evaluation function (log MAR) is expressed by the followingequation (2) which is an approximate equation: logMAR=X×log₁₀2×{curvature of field²+(residual astigmatism/2)²}^(1/2)   (2)3. A spectacle lens designing method according to clam 1, wherein saidmerit function includes an evaluation function on a distortionaberration (residual distortion aberration DIST) and said evaluationfunction is expressed by the following equation (3): residual distortionaberration DIST=Sign×100×(absolute value of residual visual anglemagnification/absolute value of central visual angle magnification M ⁰ )  (3) where: the residual visual angle magnification is the distortionaberration defined from the spectacle lens designing in which theListing's Law is taken into consideration; and Sign is apositive/negative sign.
 4. A spectacle lens designing method accordingto any one of claim 1 to claim 3, wherein said merit function is used inoptimization calculation of lens designing of a bi-aspherical lens inwhich a front surface is an axially symmetrical aspherical surface and arear surface is an aspherical surface expressed by the followingequation (4): $\begin{matrix}{{Z2} = {{c(\theta)} \cdot {r^{2}/\left( {1 + \sqrt{\left. {1 - {\left( {1 + {k(\theta)}} \right) \cdot {c(\theta)}^{2} \cdot r^{2}}} \right)} + {\sum\limits_{n}{{a\left( {n,\theta} \right)} \cdot r^{n}}}} \right.}}} & (4)\end{matrix}$

where: c(θ), k(θ) are functions for an azimuth θ; a(n, θ) is a functionfor an n degree of a distance r and the azimuth θ; as for a definitiondomain of the azimuth θ, 0 degree to 90 degrees represents 0 degree to360 degrees due to plane symmetry of an astigmatic lens; c(θ) is acurvature of a lens center and is expressed by the following equation(5) based on an Euler's theorem, letting a curvature of a spectacleprincipal meridian in a Gaussian curve theorem be c(0) at 0 degree andc(90) at 90 degrees. In this case, 0 degree is a spherical diopter axisand 90 degrees is a cylinder diopter axis; c(θ)=c(0)·cos² θ+c(90)·sin²θ  (5) k(θ), which is similar to said c(θ), represents an equation inwhich the sign c is replaced by the sign k in said equation (5); anda(n, θ) satisfies requirements of plane continuity and plane symmetry,is a surface further satisfying a requirement of a surface which iscapable of controlling an aberration due to an angle deviation whichoccurs due to the Listing's Law, and further satisfies the followingconditions {circle over (1)} to {circle over (4)}: {circle over (1)}:having a functional relation of the azimuth θ from 0 degree to 90degrees; {circle over (2)}: a linear differential coefficient of theazimuth θ is 0 from 0 degree to 90 degrees; {circle over (3)}: a highdegree differential coefficient is continuous; and {circle over (4)}:having a control parameter group Ps(n) which is capable of controlling avalue of a(n, θ) at an angle θ of a function between the azimuths of 0degree and 90 degrees (where 1 to 3 are preferable for the number of s,and n signifies a degree in said equation (4)).
 5. A spectacle lensdesigning method according to claim 4, wherein a(n, θ) in said equation(4) is expressed by the following equation (6) which is a quarticpolynominal of the azimuth θ, letting a be a(n, 0), a(n, 45), and a(n,90) when the azimuth θ is 0 degree, 45 degrees, and 90 degreesrespectively: a(n, θ)=a(n, 0)+(−11·a(n, 0)+16·a(n, 45)−5·a(n,90))·θ²/(4·90²)+(9·a(n, 0)−16·a(n, 45)+7·a(n, 90))·θ³/(4·90³)+(−2·a(n,0)+4·a(n, 45)−2·a(n, 90))·θ⁴/(4·90⁴)   (6) where a control parameter isone for the degree n of the distance r from the center and a controlparameter P1(n) is a(n, 45).
 6. A spectacle lens designing methodaccording to claim 4, wherein a(n, θ) in said equation (4) is expressedby the following equation (7), letting a be a(n, 0) and a(n, 90) whenthe azimuth θ is 0 degree and 90 degrees respectively: a(n, θ)=a(n,0)·cos² θ+a(n, 90)·sin² θ+P1(n)·sin² (2·θ)   (7) where a controlparameter is one control parameter P1(n) for the degree n of thedistance r from the center.
 7. A spectacle lens, being designed by thespectacle lens designing method according to any one of claim 1 to claim6.